SIMP OPERATIONS AND THE STRANGE BUT BEAUTIFUL JANUS-FACED PC MICHAEL SICILIANO HIS PAPER EXPLORES some of the logical implications of the use of transposition and inversion operations defined on the set of pcs. Specifically, those operations bring with them a hodgepodge of associated SIMP operations that incompletely and incompatibly describe some, but not all, situations in which some pcs move clockwise around the pc-circle while others move counterclockwise the same number of places. Many of these SIMP operations have been described before, independently of each other, and their musical applications have been explored. I will provide an extensive analysis of passages from Bartók’s Suite, op. 14, to illustrate one set of SIMP operations that has not been previously discussed, and otherwise refer the reader to published analyses that demonstrate the usefulness of the other sets. That analysis will also suggest the desirability of making T 80 Perspectives of New Music these operations compatible, and the impossibility of doing so using just pcs. I will suggest a way of making all these operations compatible, and I will end by illustrating the usefulness of this approach with an analysis. The way of making them compatible does not, of course, arise from a logical implication of inversion of pcs. However, it is strongly suggested by the use of inversion with many musical objects that are not inversionally symmetric, and can perhaps be understood to make the mathematical model of inversion of pcs more accurately reflect the analytic practice of asserting an inversional relation among pcs.1 Because some of the logical implications of inversion become counter-intuitive, I will start by belaboring a simpler system, one with only transpositions. Imagine a passage in which a melody {C, D, E, F} is accompanied by a chord {Eb, F, G, Ab}. I might propose the transformation T3, transposition by three semi-tones clockwise, to describe the relation between the melody and its accompaniment. At this point, I have committed myself to nothing more than the existence of T3. If I wish to add to T3 the idea that operations combine to form other operations, then T3 and this idea of combination commit me to a system that includes T0, T3, T6, and T9. This commitment exists whether the music at hand actually makes use of all four operations. It is worth observing that T3 and the idea of combination do not commit me to the system of twelve Tns, but only to these four of them. Familiarly, the idea of combination and T1 (or T5, T7, or T11) do commit me to all twelve Tns, again, whether or not the music at hand engages all twelve. Say instead that the melody {C, D, E, F} is accompanied by the chord {C#, B, A, G#}, suggesting the transformation I1. I1 and the idea of combination commit me only to I1 and T0 (the identity operation).2 Inversions, of course, are their own inverses, so any inversion (and the idea of combination) commits me only to that inversion and T0. However, inversions combine with transpositions to form other inversions . Therefore, if I am committed to the twelve Tns and to a single In, I am committed to the entire Tn/In group of twelve transpositions and twelve inversions, again, whether or not the music at hand engages all twenty-four transformations. Indeed, accepting any inversion, and just T1 (or T5, T7, or T11) and the idea of combination, commits me to the full group of twelve Tns and twelve Ins.3 So far I have been belaboring simple systems. The next step is less familiar. In being committed to the Tn/In group, I must also be committed to everything logically implied by that group. The group contains many subsets (some of which are sub-groups), and therefore whatever is implied by any of those subsets is also implied by the group that contains them. That is, if I have a group, I thereby also already SIMP Operations 81 have all of its sub-groups, and thereby I also have everything that comes along with those sub-groups. Some of the sub-groups of the Tn/In group do bring baggage along with...
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